November  2017, 22(9): 3529-3545. doi: 10.3934/dcdsb.2017178

A preconditioned fast Hermite finite element method for space-fractional diffusion equations

1. 

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

2. 

Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

* Corresponding author: Aijie Cheng

Received  August 2016 Revised  April 2017 Published  July 2017

We develop a fast Hermite finite element method for a one-dimensional space-fractional diffusion equation, by proving that the stiffness matrix of the method can be expressed as a Toeplitz block matrix. Then a block circulant preconditioner is presented. Numerical results are presented to show the utility of the fast method.

Citation: Meng Zhao, Aijie Cheng, Hong Wang. A preconditioned fast Hermite finite element method for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3529-3545. doi: 10.3934/dcdsb.2017178
References:
[1]

T. Basu and H. Wang, A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Mod., 9 (2012), 658-666.   Google Scholar

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D. BensonS. W. Wheatcraft and M. M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000), 1413-1423.  doi: 10.1029/2000WR900032.  Google Scholar

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W. BuY. Tang and J. Yang, Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276 (2014), 26-38.  doi: 10.1016/j.jcp.2014.07.023.  Google Scholar

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R. H. Chan, Circulant preconditioners for Hermitian Toeplitz systems, SIAM Journal on Matrix Analysis and Applications, 10 (1989), 542-550.  doi: 10.1137/0610039.  Google Scholar

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R. H. Chan and X. Q. Jin, A family of block preconditioners for block systems, SIAM journal on scientific and statistical computing, 13 (1992), 1218-1235.  doi: 10.1137/0913070.  Google Scholar

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P. J. Davis, Circulant Matrices, Wiley-Intersciences, New York, 1979.  Google Scholar

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W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM Journal on Numerical Analysis, 47 (2008), 204-226.  doi: 10.1137/080714130.  Google Scholar

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K. Diethelm and A. D. Freed, An efficient algorithm for the evaluation of convolution integrals, Computers and Mathematics with Applications, 51 (2006), 204-226.  doi: 10.1016/j.camwa.2005.07.010.  Google Scholar

[10]

N. Du and H. Wang, A fast finite element method for space-fractional dispersion equations on bounded domains in $\mathbb{R}^2$, SIAM J. Sci. Comput., 37 (2015), A1614-A1635.  doi: 10.1137/15M1007458.  Google Scholar

[11]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods for Partial Differential Equations, 22 (2005), 558-576.  doi: 10.1002/num.20112.  Google Scholar

[12]

V. J. Ervin and J. P. Roop, Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in $R^{d}$, Numer. Methods for Partial Differential Equations, 23 (2007), 256-281.  doi: 10.1002/num.20169.  Google Scholar

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R. M. Gray, Toeplitz and circulant matrices: A review, Communications and Information Theory, 2 (2016), 155-239.  doi: 10.1561/0100000006.  Google Scholar

[14]

J. Jia and H. Wang, A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299 (2015), 842-862.  doi: 10.1016/j.jcp.2015.06.028.  Google Scholar

[15]

Y. Jiang and X. Xu, Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302 (2015), 374-392.  doi: 10.1016/j.jcp.2015.08.052.  Google Scholar

[16]

S. L. Lei and H. W. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715-725.  doi: 10.1016/j.jcp.2013.02.025.  Google Scholar

[17]

C. LiZ. Zhao and Y. Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Computers and Mathematics with Applications, 62 (2011), 855-875.  doi: 10.1016/j.camwa.2011.02.045.  Google Scholar

[18]

X. Li and C. Xu, The existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016-1051.  doi: 10.4208/cicp.020709.221209a.  Google Scholar

[19]

F. R. LinS. W. Yang and X. Q. Jin, Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256 (2014), 109-117.  doi: 10.1016/j.jcp.2013.07.040.  Google Scholar

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F. LiuV. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2014), 209-219.  doi: 10.1016/j.cam.2003.09.028.  Google Scholar

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C.~W. Lv and C. Xu, Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations, Int. J. Numer. Anal. Mod., 12 (2015), 384-400.   Google Scholar

[23]

M. M. MeerschaertH. P. Scheffler and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), 249-261.  doi: 10.1016/j.jcp.2005.05.017.  Google Scholar

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M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.  Google Scholar

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R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

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H. G. SunW. Chen and Y. Q. Chen, Fractional differential models for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 389 (2010), 2719-2724.  doi: 10.1016/j.physa.2010.02.030.  Google Scholar

[30]

H. TianH. Wang and W. Q. Wang, An efficient collocation method for a non-local diffusion model, Int. J. Numer. Anal. Mod., 10 (2013), 815-825.   Google Scholar

[31]

P. Vabishchevich, Numerical solution of nonstationary problems for a convection and a space-fractional diffusion equation, Int. J. Numer. Anal. Mod., 13 (2016), 296-309.   Google Scholar

[32]

H. Wang and D. P. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM Journal on Numerical Analysis, 51 (2013), 1088-1107.  doi: 10.1137/120892295.  Google Scholar

[33]

H. WangD. P. Yang and S. F. Zhu, Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, Journal of Scientific Computing, 70 (2017), 429-449.  doi: 10.1007/s10915-016-0196-7.  Google Scholar

[34]

H. Wang and X. H. Zhang, A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Comput. Phys., 281 (2015), 67-81.  doi: 10.1016/j.jcp.2014.10.018.  Google Scholar

[35]

H. Wang and T. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444-A2458.  doi: 10.1137/12086491X.  Google Scholar

[36]

H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49-57.  doi: 10.1016/j.jcp.2012.07.045.  Google Scholar

[37]

H. WangK. Wang and T. Sircar, A direct $O(N\log^2 N)$ finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104.  doi: 10.1016/j.jcp.2010.07.011.  Google Scholar

[38]

L. L. WeiY. N. He and Y. Zhang, Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method, Int. J. Numer. Anal. Mod., 10 (2013), 430-444.   Google Scholar

show all references

References:
[1]

T. Basu and H. Wang, A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Mod., 9 (2012), 658-666.   Google Scholar

[2]

D. BensonS. W. Wheatcraft and M. M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000), 1413-1423.  doi: 10.1029/2000WR900032.  Google Scholar

[3]

W. BuY. Tang and J. Yang, Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276 (2014), 26-38.  doi: 10.1016/j.jcp.2014.07.023.  Google Scholar

[4]

R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), 427-482.  doi: 10.1137/S0036144594276474.  Google Scholar

[5]

R. H. Chan, Circulant preconditioners for Hermitian Toeplitz systems, SIAM Journal on Matrix Analysis and Applications, 10 (1989), 542-550.  doi: 10.1137/0610039.  Google Scholar

[6]

R. H. Chan and X. Q. Jin, A family of block preconditioners for block systems, SIAM journal on scientific and statistical computing, 13 (1992), 1218-1235.  doi: 10.1137/0913070.  Google Scholar

[7]

P. J. Davis, Circulant Matrices, Wiley-Intersciences, New York, 1979.  Google Scholar

[8]

W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM Journal on Numerical Analysis, 47 (2008), 204-226.  doi: 10.1137/080714130.  Google Scholar

[9]

K. Diethelm and A. D. Freed, An efficient algorithm for the evaluation of convolution integrals, Computers and Mathematics with Applications, 51 (2006), 204-226.  doi: 10.1016/j.camwa.2005.07.010.  Google Scholar

[10]

N. Du and H. Wang, A fast finite element method for space-fractional dispersion equations on bounded domains in $\mathbb{R}^2$, SIAM J. Sci. Comput., 37 (2015), A1614-A1635.  doi: 10.1137/15M1007458.  Google Scholar

[11]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods for Partial Differential Equations, 22 (2005), 558-576.  doi: 10.1002/num.20112.  Google Scholar

[12]

V. J. Ervin and J. P. Roop, Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in $R^{d}$, Numer. Methods for Partial Differential Equations, 23 (2007), 256-281.  doi: 10.1002/num.20169.  Google Scholar

[13]

R. M. Gray, Toeplitz and circulant matrices: A review, Communications and Information Theory, 2 (2016), 155-239.  doi: 10.1561/0100000006.  Google Scholar

[14]

J. Jia and H. Wang, A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299 (2015), 842-862.  doi: 10.1016/j.jcp.2015.06.028.  Google Scholar

[15]

Y. Jiang and X. Xu, Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302 (2015), 374-392.  doi: 10.1016/j.jcp.2015.08.052.  Google Scholar

[16]

S. L. Lei and H. W. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715-725.  doi: 10.1016/j.jcp.2013.02.025.  Google Scholar

[17]

C. LiZ. Zhao and Y. Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Computers and Mathematics with Applications, 62 (2011), 855-875.  doi: 10.1016/j.camwa.2011.02.045.  Google Scholar

[18]

X. Li and C. Xu, The existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016-1051.  doi: 10.4208/cicp.020709.221209a.  Google Scholar

[19]

F. R. LinS. W. Yang and X. Q. Jin, Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256 (2014), 109-117.  doi: 10.1016/j.jcp.2013.07.040.  Google Scholar

[20]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.  Google Scholar

[21]

F. LiuV. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2014), 209-219.  doi: 10.1016/j.cam.2003.09.028.  Google Scholar

[22]

C.~W. Lv and C. Xu, Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations, Int. J. Numer. Anal. Mod., 12 (2015), 384-400.   Google Scholar

[23]

M. M. MeerschaertH. P. Scheffler and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), 249-261.  doi: 10.1016/j.jcp.2005.05.017.  Google Scholar

[24]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.  Google Scholar

[25]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[26]

H. K. Pang and H. W. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703.  doi: 10.1016/j.jcp.2011.10.005.  Google Scholar

[27]

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.  Google Scholar

[28]

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. , SIAM, Rhode Island, 2003. doi: 10.1137/1. 9780898718003.  Google Scholar

[29]

H. G. SunW. Chen and Y. Q. Chen, Fractional differential models for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 389 (2010), 2719-2724.  doi: 10.1016/j.physa.2010.02.030.  Google Scholar

[30]

H. TianH. Wang and W. Q. Wang, An efficient collocation method for a non-local diffusion model, Int. J. Numer. Anal. Mod., 10 (2013), 815-825.   Google Scholar

[31]

P. Vabishchevich, Numerical solution of nonstationary problems for a convection and a space-fractional diffusion equation, Int. J. Numer. Anal. Mod., 13 (2016), 296-309.   Google Scholar

[32]

H. Wang and D. P. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM Journal on Numerical Analysis, 51 (2013), 1088-1107.  doi: 10.1137/120892295.  Google Scholar

[33]

H. WangD. P. Yang and S. F. Zhu, Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, Journal of Scientific Computing, 70 (2017), 429-449.  doi: 10.1007/s10915-016-0196-7.  Google Scholar

[34]

H. Wang and X. H. Zhang, A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Comput. Phys., 281 (2015), 67-81.  doi: 10.1016/j.jcp.2014.10.018.  Google Scholar

[35]

H. Wang and T. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444-A2458.  doi: 10.1137/12086491X.  Google Scholar

[36]

H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49-57.  doi: 10.1016/j.jcp.2012.07.045.  Google Scholar

[37]

H. WangK. Wang and T. Sircar, A direct $O(N\log^2 N)$ finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104.  doi: 10.1016/j.jcp.2010.07.011.  Google Scholar

[38]

L. L. WeiY. N. He and Y. Zhang, Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method, Int. J. Numer. Anal. Mod., 10 (2013), 430-444.   Google Scholar

Table 1.  The condition number of stiffness matrix $\mathsf{A}$ with $K=1$, $\gamma=0.5$, $\beta=0.5$
h $2^{-8} $ $2^{-9} $$2^{-10} $
$Cond(\mathsf{A})$ $ 2.329917 \times 10^{6}$ $ 9.320396 \times 10^{6}$ $ 3.728232 \times 10^{7}$
h $2^{-8} $ $2^{-9} $$2^{-10} $
$Cond(\mathsf{A})$ $ 2.329917 \times 10^{6}$ $ 9.320396 \times 10^{6}$ $ 3.728232 \times 10^{7}$
Table 2.  The $L^2$ error and $H^{1-\frac{\beta}{2}}$ error of the cubic Hermite element method with $\beta=0.1,0.5,0.9$
$\beta$hDOF $ \|u_h - u \|_{L^2}$order $ \|u_h - u \|_{H^{1-\frac{\beta}{2}}}$order
0.5 $2^{-3} $16 $ 4.872892 \times 10^{-4}$ $ 2.512934 \times 10^{-3}$
$2^{-4} $32 $ 3.573631 \times 10^{-5}$3.7693 $ 2.576521 \times 10^{-4}$3.2858
$2^{-5} $64 $ 2.236217 \times 10^{-6}$3.9982 $ 2.481970 \times 10^{-5}$3.3758
$2^{-6} $128$ 1.342295 \times 10^{-7}$4.0582$ 2.249708 \times 10^{-6}$3.4636
$2^{-7} $256$ 7.909661 \times 10^{-9}$4.0849$ 2.033970 \times 10^{-7}$3.4673
0.1 $2^{-3} $16 $ 7.851432 \times 10^{-4}$ $ 8.011606 \times 10^{-3}$
$2^{-4} $32 $ 6.192597 \times 10^{-5}$3.6643 $ 1.304843 \times 10^{-3}$2.6352
$2^{-5} $64 $ 4.208159 \times 10^{-6}$3.8792 $ 1.805192 \times 10^{-4}$2.8536
$2^{-6} $128$ 2.713900 \times 10^{-7}$3.9547$ 2.295487 \times 10^{-5}$2.9752
$2^{-7} $256$ 1.794745 \times 10^{-8}$3.9185$ 2.808095 \times 10^{-6}$3.0311
0.9 $2^{-3} $16 $ 3.622149 \times 10^{-4}$ $ 4.846137 \times 10^{-4}$
$2^{-4} $32 $ 2.767086 \times 10^{-5}$3.7104 $ 4.110739 \times 10^{-5}$3.5593
$2^{-5} $64 $ 1.826287 \times 10^{-6}$3.9213 $ 3.012015 \times 10^{-6}$3.7705
$2^{-6} $128$ 1.180438 \times 10^{-7}$3.9515$ 2.132241 \times 10^{-7}$3.8202
$2^{-7} $256$ 7.376961 \times 10^{-9}$4.0001$ 1.415327 \times 10^{-8}$3.9131
$\beta$hDOF $ \|u_h - u \|_{L^2}$order $ \|u_h - u \|_{H^{1-\frac{\beta}{2}}}$order
0.5 $2^{-3} $16 $ 4.872892 \times 10^{-4}$ $ 2.512934 \times 10^{-3}$
$2^{-4} $32 $ 3.573631 \times 10^{-5}$3.7693 $ 2.576521 \times 10^{-4}$3.2858
$2^{-5} $64 $ 2.236217 \times 10^{-6}$3.9982 $ 2.481970 \times 10^{-5}$3.3758
$2^{-6} $128$ 1.342295 \times 10^{-7}$4.0582$ 2.249708 \times 10^{-6}$3.4636
$2^{-7} $256$ 7.909661 \times 10^{-9}$4.0849$ 2.033970 \times 10^{-7}$3.4673
0.1 $2^{-3} $16 $ 7.851432 \times 10^{-4}$ $ 8.011606 \times 10^{-3}$
$2^{-4} $32 $ 6.192597 \times 10^{-5}$3.6643 $ 1.304843 \times 10^{-3}$2.6352
$2^{-5} $64 $ 4.208159 \times 10^{-6}$3.8792 $ 1.805192 \times 10^{-4}$2.8536
$2^{-6} $128$ 2.713900 \times 10^{-7}$3.9547$ 2.295487 \times 10^{-5}$2.9752
$2^{-7} $256$ 1.794745 \times 10^{-8}$3.9185$ 2.808095 \times 10^{-6}$3.0311
0.9 $2^{-3} $16 $ 3.622149 \times 10^{-4}$ $ 4.846137 \times 10^{-4}$
$2^{-4} $32 $ 2.767086 \times 10^{-5}$3.7104 $ 4.110739 \times 10^{-5}$3.5593
$2^{-5} $64 $ 1.826287 \times 10^{-6}$3.9213 $ 3.012015 \times 10^{-6}$3.7705
$2^{-6} $128$ 1.180438 \times 10^{-7}$3.9515$ 2.132241 \times 10^{-7}$3.8202
$2^{-7} $256$ 7.376961 \times 10^{-9}$4.0001$ 1.415327 \times 10^{-8}$3.9131
Table 3.  Performance of Gauss, CGS methods with $\beta=0.5$
GaussCGS
hCPU(s)Itr. # CPU(s)Itr. #
$2^{-6}$0.09160.2103137
$2^{-7}$0.48651.5369272
$2^{-8}$3.58589.8734415
$2^{-9}$24.477462.7682665
$2^{-10}$186.9874391.5595899
$2^{-11}$1485.14502731.07511676
$2^{-12}$out of memoryout of memory
GaussCGS
hCPU(s)Itr. # CPU(s)Itr. #
$2^{-6}$0.09160.2103137
$2^{-7}$0.48651.5369272
$2^{-8}$3.58589.8734415
$2^{-9}$24.477462.7682665
$2^{-10}$186.9874391.5595899
$2^{-11}$1485.14502731.07511676
$2^{-12}$out of memoryout of memory
Table 4.  Performance of FCGS, SFCGS and PFCGS methods with $\beta=0.5$
FCGSSFCGSCFCGS
hCPU(s)Itr. # CPU(s)Itr. # CPU(s)Itr. #
$2^{-6}$0.08321370.030570.03419
$2^{-7}$0.18092720.035870.041910
$2^{-8}$0.28904150.050370.044511
$2^{-9}$0.99076650.068080.065312
$2^{-10}$2.45258990.093280.104614
$2^{-11}$16.075216760.153690.242216
$2^{-12}$26.475164210.191490.595519
$2^{-13}$N/A $>$ 30,0000.6319101.410722
FCGSSFCGSCFCGS
hCPU(s)Itr. # CPU(s)Itr. # CPU(s)Itr. #
$2^{-6}$0.08321370.030570.03419
$2^{-7}$0.18092720.035870.041910
$2^{-8}$0.28904150.050370.044511
$2^{-9}$0.99076650.068080.065312
$2^{-10}$2.45258990.093280.104614
$2^{-11}$16.075216760.153690.242216
$2^{-12}$26.475164210.191490.595519
$2^{-13}$N/A $>$ 30,0000.6319101.410722
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